Question: You have found the following ages (in years) of all 5 meerkats at your local zoo: $ 11,\enspace 5,\enspace 11,\enspace 1,\enspace 1$ What is the average age of the meerkats at your zoo? What is the standard deviation? You may round your answers to the nearest tenth.
Solution: Because we have data for all 5 meerkats at the zoo, we are able to calculate the population mean $({\mu})$ and population standard deviation $({\sigma})$ To find the population mean , add up the values of all $5$ ages and divide by $5$ $ {\mu} = \dfrac{\sum\limits_{i=1}^{{N}} x_i}{{N}} = \dfrac{\sum\limits_{i=1}^{{5}} x_i}{{5}} $ $ {\mu} = \dfrac{11 + 5 + 11 + 1 + 1}{{5}} = {5.8\text{ years old}} $ Find the squared deviations from the mean for each meerkat. Age $x_i$ Distance from the mean $(x_i - {\mu})$ $(x_i - {\mu})^2$ $11$ years $5.2$ years $27.04$ years $^2$ $5$ years $-0.8$ years $0.64$ years $^2$ $11$ years $5.2$ years $27.04$ years $^2$ $1$ year $-4.8$ years $23.04$ years $^2$ $1$ year $-4.8$ years $23.04$ years $^2$ Because we used the population mean $({\mu})$ to compute the squared deviations from the mean , we can find the variance $({\sigma^2})$ , without introducing any bias, by simply averaging the squared deviations from the mean $ {\sigma^2} = \dfrac{\sum\limits_{i=1}^{{N}} (x_i - {\mu})^2}{{N}} $ $ {\sigma^2} = \dfrac{{27.04} + {0.64} + {27.04} + {23.04} + {23.04}} {{5}} $ $ {\sigma^2} = \dfrac{{100.8}}{{5}} = {20.16\text{ years}^2} $ As you might guess from the notation, the population standard deviation $({\sigma})$ is found by taking the square root of the population variance $({\sigma^2})$ ${\sigma} = \sqrt{{\sigma^2}}$ $ {\sigma} = \sqrt{{20.16\text{ years}^2}} = {4.5\text{ years}} $ The average meerkat at the zoo is 5.8 years old. There is a standard deviation of 4.5 years.